Discrete-time Martingale Research Articles

Bubbles in discrete-time models

We introduce a new definition of bubbles in discrete-time models based on the discounted stock price losing mass under an equivalent martingale measure at some finite drawdown. We provide equivalent probabilistic characterisations of this definition and give examples of discrete-time martingales that are bubbles and others that are not. In the Markovian case, we provide sufficient analytic conditions for the presence of bubbles. We also show that the existence of bubbles is directly linked to the existence of a non-trivial solution to a linear Volterra integral equation of the second kind involving the Markov kernel. Finally, we show that our definition of bubbles in discrete time is consistent with the strict local martingale definition of bubbles in continuous time in the sense that a properly discretised strict local martingale in continuous time is a bubble in discrete time.

Fractional integration for irregular martingales

We suggest two versions of the Hardy--Littlewood--Sobolev inequality for discrete time martingales. In one version, the fractional integration operator is a martingale transform, however, it may vanish if the filtration is excessively irregular; the second version lacks the martingale property while being analytically meaningful for an arbitrary filtration.

The potential of the shadow measure

It is well known that given two probability measures μ and ν on R in convex order there exists a discrete-time martingale with these marginals. Several solutions are known (for example from the literature on the Skorokhod embedding problem in Brownian motion). But, if we add a requirement that the martingale should minimise the expected value of some functional of its starting and finishing positions then the problem becomes more difficult. Beiglböck and Juillet (Ann. Probab. 44 (2016) 42–106) introduced the shadow measure which induces a family of martingale couplings, and solves the optimal martingale transport problem for a class of bivariate objective functions. In this article we extend their (existence and uniqueness) results by providing an explicit construction of the shadow measure and, as an application, give a simple proof of its associativity.

On a discrete fractional stochastic Grönwall inequality and its application in the numerical analysis of stochastic FDEs involving a martingale

Abstract The aim of this paper is to derive a novel discrete form of stochastic fractional Grönwall lemma involving a martingale. The proof of the derived inequality is accomplished by a corresponding no randomness form of the discrete fractional Grönwall inequality and an upper bound for discrete-time martingales representing the supremum in terms of the infimum. The release of a martingale term on the right-hand side of the given inequality and the graded L1 difference formula for the time Caputo fractional derivative of order 0 < α < 1 on the left-hand side are the main challenges of the stated and proved main theorem. As an example of application, the constructed theorem is used to derive an a priori estimate for a discrete stochastic fractional model at the end of the paper.

Discrete fractional stochastic Grönwall inequalities arising in the numerical analysis of multi-term fractional order stochastic differential equations

This paper is devoted to the rigorous derivation of some discrete versions of stochastic Grönwall inequalities involving a martingale, which are commonly used in the numerical analysis of multi-term stochastic time-fractional diffusion equations. A Grönwall lemma is also established to deal with the numerical analysis of multi-term stochastic fractional diffusion equations with delay. The proofs of the established inequalities are based on a corresponding deterministic version of the discrete fractional Grönwall lemma in case of smooth solutions and an inequality bounding the supremum in terms of the infimum for discrete time martingales. A numerical application is introduced finally in which the constructed inequalities are handled to derive a priori estimates for a discrete fractional stochastic model.

On Bernstein type inequalities for stochastic integrals of multivariate point processes

In this paper, we first obtain a Bernstein type of concentration inequality for stochastic integrals of multivariate point processes under some conditions through the Doléans-Dade exponential formula, and then derive a uniform exponential inequality using a generic chaining argument. As a direct consequence, we obtain an upper bound for a sequence of discrete time martingales indexed by a class of functionals. Finally, we apply the uniform exponential bound to nonparametric maximum likelihood estimators and provide a rate of convergence in terms of Hellinger distance, which is an improvement of earlier work of van de Geer (1995).

On quantitative bounds in the mean martingale central limit theorem

We provide explicit bounds on the Wasserstein distance between discrete time martingales and the standard normal distribution. The proofs are based on a combination of Lindeberg’s and Stein’s method.

Local martingales in discrete time

For any discrete-time $\mathsf{P} $–local martingale $S$ there exists a probability measure $\mathsf{Q} \sim \mathsf{P} $ such that $S$ is a $\mathsf{Q} $–martingale. A new proof for this result is provided. The core idea relies on an appropriate modification of an argument by Chris Rogers, used to prove a version of the fundamental theorem of asset pricing in discrete time. This proof also yields that, for any $\varepsilon >0$, the measure $\mathsf{Q} $ can be chosen so that $\frac{\mathrm {d} \mathsf {Q}} {\mathrm{d} \mathsf{P} } \leq 1+\varepsilon $.

On one property of martingales with conditionally Gaussian increments and its application in the theory of nonasymptotic inference

A transformation of a discrete-time martingale with conditionally Gaussian increments into a sequence of i.i.d. standard Gaussian random variables is proposed as based on a sequence of stopping times constructed using the quadratic variation. It is shown that sequential estimators for the parameters in AR(1) and generalized first-order autoregressive models have a nonasymptotic normal distribution.

A discrete stochastic Gronwall lemma

The purpose of this paper is the derivation of a discrete version of the stochastic Gronwall lemma involving a martingale. The proof is based on a corresponding deterministic version of the discrete Gronwall lemma and an inequality bounding the supremum in terms of the infimum for discrete time martingales. As an application the proof of an a priori estimate for the backward Euler–Maruyama method is included.

Benchmarking in two price financial markets

No arbitrage for two price economies with no locally risk free asset implies that suitably benchmarked prices are nonlinear martingales. However, both the benchmarking asset and the measure change depend on the process being benchmarked. Further assumptions allow the nonlinear martingales in discrete time to become expectations with respect to a nonadditivity probability. Such nonlinear expectations are imminently reasonable given the lack of experience with tail events on both sides of the gain loss spectrum. Continuous time extensions employ measure distortions. The general valuation of economic activities and the leveraging of stability in benchmarked price processes is then addressed. Traditional asset pricing questions and investigations are then reopened for benchmarked prices. In particular, the analytics for benchmarked option pricing and the asset pricing theory for benchmarked prices in a limiting stationary state are developed.

Scale-Free Property for Degrees and Weights in an N-Interactions Random Graph Model*

A general random graph evolution mechanism is defined. The evolution is based on the interactions of N vertices. Besides the interactions of the new vertex and the old ones, interactions among old vertices are also allowed. Moreover, both preferential attachment and uniform choice are possible. A vertex in the graph is characterized by its degree and its weight. The weight of a given vertex is the number of interactions of the vertex. The asymptotic behavior of the graph is studied. Scale-free properties both for the degrees and the weights are proved. It turns out that any exponent in (2,∞) can be achieved. The proofs are based on discrete time martingale theory.

Relativities in Financial Markets

No arbitrage for two price economies with no locally risk free asset implies that suitably deflated prices are nonlinear martingales. However, both the deflating process and the measure change depend on the process being deflated. Further assumptions allow the nonlinear martingales in discrete time to become expectations with respect to a nonadditive probability. Such nonlinear expectations that distort probabilities are imminently reasonable given the lack of experience with tail events on both sides of the gain loss spectrum. Continuous time extensions replace probability distortions with measure distortions that distort the arrival rates of jumps. The general valuation of economic activities and the leveraging of stability in deflated price processes is then addressed. Applications include the pricing of options on relativities and the asset pricing theory for relativities in a limiting stationary state.

A representation theorem for smooth Brownian martingales

We show that, under certain smoothness conditions, a Brownian martingale, when evaluated at a fixed time, can be represented via an exponential formula at a later time. The time-dependent generator of this exponential operator only depends on the second order Malliavin derivative operator evaluated along a ‘frozen path’. The exponential operator can be expanded explicitly to a series representation, which resembles the Dyson series of quantum mechanics. Our continuous-time martingale representation result can be proven independently by two different methods. In the first method, one constructs a time-evolution equation, by passage to the limit of a special case of a backward Taylor expansion of an approximating discrete-time martingale. The exponential formula is a solution of the time-evolution equation, but we emphasize in our article that the time-evolution equation is a separate result of independent interest. In the second method, we use the property of denseness of exponential functions. We provide several applications of the exponential formula, and briefly highlight numerical applications of the backward Taylor expansion.

Weak approximation of second-order BSDEs

We study the weak approximation of the second-order backward SDEs (2BSDEs), when the continuous driving martingales are approximated by discrete time martingales. We establish a convergence result for a class of 2BSDEs, using both robustness properties of BSDEs, as proved in Briand, Delyon and Mémin [ Stochastic Process. Appl. 97 (2002) 229–253], and tightness of solutions to discrete time BSDEs. In particular, when the approximating martingales are given by some particular controlled Markov chains, we obtain several concrete numerical schemes for 2BSDEs, which we illustrate on specific examples.

Weights of Cliques in a Random Graph Model Based on Three-Interactions*

We consider a random graph evolution rule. The graph evolution is based on interactions of three vertices. The weight of a clique is the number of its interactions. We describe the asymptotic behavior of the weights. It is known that the weight distribution of the vertices is asymptotically a power law. Here we prove that the weight distributions both of the edges and of the triangles are also asymptotically power laws. The proofs are based on discrete-time martingale methods. Some numerical results are also presented.

Quadratic variation of martingales in Riesz spaces

We derive quadratic variation inequalities for discrete-time martingales, sub- and supermartingales in the measure-free setting of Riesz spaces. Our main result is a Riesz space analogue of Austinʼs sample function theorem, on convergence of the quadratic variation processes of martingales.

On the rate of convergence in the martingale central limit theorem

Consider a discrete-time martingale, and let $V^2$ be its normalized quadratic variation. As $V^2$ approaches 1, and provided that some Lindeberg condition is satisfied, the distribution of the rescaled martingale approaches the Gaussian distribution. For any $p\geq 1$, (Ann. Probab. 16 (1988) 275-299) gave a bound on the rate of convergence in this central limit theorem that is the sum of two terms, say $A_p+B_p$, where up to a constant, $A_p={\|V^2-1\|}_p^{p/(2p+1)}$. Here we discuss the optimality of this term, focusing on the restricted class of martingales with bounded increments. In this context, (Ann. Probab. 10 (1982) 672-688) sketched a strategy to prove optimality for $p=1$. Here we extend this strategy to any $p\geq 1$, thereby justifying the optimality of the term $A_p$. As a necessary step, we also provide a new bound on the rate of convergence in the central limit theorem for martingales with bounded increments that improves on the term $B_p$, generalizing another result of (Ann. Probab. 10 (1982) 672-688).

Numerical schemes for G-Expectations

We consider a discrete time analog of $G$--expectations and we prove that in the case where the time step goes to 0 the corresponding values converge to the original $G$--expectation. Furthermore we provide error estimates for the convergence rate. This paper is continuation of [4]. Our main tool is a strong approximation theorem which we derive for general discrete time martingales.

On the Dirichlet problem

A particular case of the Dirichlet problem is solved using the Convergence Theorem for discrete-time martingales and the mean value property of harmonic functions as the main tools.